Our results for the exponent in continuum only and line blanketed convective atmospheres are very similar to each other. The dependence of for these models is mostly on the effective temperature, being very weak, albeit systematic, its dependence on the mixing length parameter (i.e. on the efficiency of convection in the energy transport).
For lower , convection becomes very efficient due to the higher density caused by the lower opacity. The entropy jump at the surface becomes small and its changes even smaller (the convective zone reaches layers closer to the surface), yielding small . For increasingly higher , the top convective layers start deeper and deeper, increasing the entropy jump and its changes at the surface and, consequently, . At high , convection loses importance altogether, and the effect weakens again, although the adiabat still is a constraint at the bottom. In between these limits reaches a maximum at 5 000 K.
Equation (1) was valid to a high degree in all the models. The values were determined by linear regression with correlation coefficients very close to 1.0: the smallest value was =0.994 for some models with =2.0. The correlation coefficients for the values calculated by using either =0.30 or 1.5 (see Table 3) were essentially equal.
3.1. Comparison with empirical determinations
Figure 3 shows our results for and empirical determinations found in the literature for the exponent. The error bars are given for the exponent only, due to lack of information on the errors in most of the works used. However, we may assume that these errors are at least K, a too optimistic figure for works using UBV, but reasonable for those using uvby or DDO photometry.
Lucy's (1967) =0.32 is a reasonable approximation for the data in Fig. 3. Our theoretical result agrees with Lucy's value in the temperature range he used and matches most of the empirical values for detached systems in Fig. 3, with only one determination showing a large discrepancy. Although the values for semi-detached and contact systems show a larger scattering, the theoretical curve still is a good approximation. One reason for the large scattering of the empirical values for the more deformed systems may be the large numerical correlation in the EB models between the exponent and the geometric parameters (inclination and star sizes) and, in some configurations, with the effective temperatures, too. Note that the above mentioned correlation has meaning exclusively for the empirical determinations of ; the theoretical values depend only on the atmosphere model parameters.
Contact and semi-detached EB should be better represented by illuminated atmospheres, since their components are closer to each other. Some calculations were made with an atmosphere grey model of =4 611 K heated by sources with =4 500 K and K, following Vaz & Nordlund (1985). The amount of incident flux may be changed through the value of , the ratio between the source star radius and the distance from its centre to the point at the surface of the reflecting star. The results (Table 4) show that the external heating increases the values roughly in proportion to the amount of the incident flux. It would be then expected that contact and semi-detached systems would lie above our polynomial fit computed for non-illuminated atmospheres. The error estimations for in Fig. 3 may be too optimistic, due to the correlations mentioned above and to the intrinsic difficulties in its empirical determinations. Better determinations of are needed to confirm observationally the theoretically expected influence of external illumination 1
The system CC Com (Ruciski 1976) has the only observational determination of for . W UMa (Eaton et al. 1980) has both components shown in Fig. 3 with by Hutchings & Hill (1973) and presents the lowest value amongst all the systems. Our model cannot reproduce the =0 predicted by Anderson & Shu (1977) for 6 400 K. The reason may be the fact that our method is unable to detect any horizontal entropy gradients, which are possibly significant in contact systems, but not in the (semi) detached ones. Our results, then, must be taken with care if applyed to contact systems, but should be valid for the detached and semi-detached ones.
3.2. One first observational test
Equation (2), with the non-grey coefficients of Table 2, was implemented in the modified version of the WD model (Wilson & Devinney 1971, Wilson 1993) described by Vaz et al. (1995). The triple PMS eclipsing system TY CrA (Casey et al. 1993, 1995) has its eclipsing secondary component early in the PMS evolutionary phase, at the very end of the Hayashi track. The light curves were analysed (Casey et al. 1997) with the WD model using both Lucy's value and our approximation for . For the secondary's effective temperature (4 800 K) our value is 0.41, about 28% larger than Lucy's value. The results obtained with our were more consistent than when using =0.32, yielding a (B8-9 ZAMS) primary star with absolute dimensions closer to what theoretically expected for a star in such phase. The reason was the increase of surface brightness in the secondary polar regions, requiring a lower inclination to reproduce the minimum depth and, consequently, larger stellar radii to reproduce the minima duration. Even though this result is encouraging, TY CrA is not, however, the most adequate system for such a test, as it is in a problematic evolutionary phase. We are searching for more favorable EBS for a definite observational test of the impact of the proposed theoretical values for on EBLC analysis, whose results will be published elsewhere.
© European Southern Observatory (ESO) 1997
Online publication: April 20, 1998